# Is the passage describing an approximation of pi or is it one of the miracles of Bais Hamikdash?

In Sepher M'lakhim we read how different measurement was taken. I wonder if this passage (Ⅰ M'lakhim 7:23) is describing an approximation of PI (3.1415...), or was this one of the hidden miracles?

וַיַּעַשׂ אֶת־הַיָּם מוּצָק עֶשֶׂר בָּאַמָּה מִשְּׂפָתֹו עַד־שְׂפָתֹו עָגֹל ׀ סָבִיב וְחָמֵשׁ בָּאַמָּה קֹומָתֹו [וּקְוֵה כ] (וְקָו ק) שְׁלֹשִׁים בָּאַמָּה יָסֹב אֹתֹו סָבִיב׃

Now he made the sea of cast metal ten cubits from brim to brim, circular in form, and its height was five cubits, and thirty cubits in circumference.

• can you define what you mean "one of the hidden miracles"? are you saying that it was a miracle that it was only 30 cubits, instead of the true value it should have been mathematically? Commented Jun 6, 2012 at 22:48
• I'm no believer in biblical literacy, but I've never seen the problem here. It doesn't say "10 cubits to infinity decimal places", it says "5, 10 and 30 cubits" That, to me, suggests that I probably rounded up from nine and a half -- why is that a problem? Commented Jun 12, 2012 at 14:30
• Thank you all for taking your time answering my questions. It is greatly appreciated.
– Ben
Commented Jun 17, 2012 at 15:40
• Inner diameter; outer circumference? Or a curved in lip on the bowl with the circumference measured around the middle of the bowl, and the diameter across the top? Frankly, I've never seen the problem here.
– TRiG
Commented Jul 17, 2012 at 18:05
• Related: judaism.stackexchange.com/q/18903.
– msh210
Commented Sep 4, 2012 at 3:32

It's approximating π, as is clear from the g'mara (Eruvin 14:1).

The problem is that that g'mara seems to be saying that it's a pretty precise approximation, and we know it's not. (Tosafos there raise this question and offer no answer.)

But to answer your question, whether it's an approximation of π or a miracle, it's the former.

• Thank you very much. I had hope for a definite answer. I'm satisfied to know that is is non. I'm satisfied with the explanation you gave. I hope that I can learn form g'mara my self some day.
– Ben
Commented Jun 6, 2012 at 16:41
• To your question, whether the verse is approximating π or describing (no pun intended) a miracle, I think there is a definite answer: it's approximating π.
– msh210
Commented Jun 6, 2012 at 18:46
• IIRC, one of the other tosafos, (Tosafos HaRosh, I think, but I can't find a copy online) rereads the gemara so that it makes more sense. I don't really remember how, though.
– jake
Commented Jun 6, 2012 at 19:32
• @jake You remember correctly. The Tosafos HaRosh explains that the gemara knew that pi is not 3, and was asking from where we know that 3 is a halachically acceptable estimation of pi (as the mishna is clear that it uses this approximation). Commented May 5, 2017 at 4:10
• @Y e z Aruch HaShulchan reads similarly to the Tosfos HaRosh, but flips it around; rather than asking how we know we can rely on π=3, he asks how we know that, since π!=3, a cylinder with a circumference of 3 is enough for the koreh of a mavoi even though that means its diameter will be less than 1. Commented Dec 19, 2018 at 13:58

The GR"A points out the following:

The word circumference (kav) is spelled קוה but pronounced קו. The gematria of the former is 111 and the latter is 106. The ratio of 111 to 106, multiplied by the approximation of 3, gives you:

(111 / 106) * 3 = 3.1415

Perhaps pi to five digits is a better approximation than 3?

• So you're taking the approximation side, not the miracle side. Commented Jun 6, 2012 at 19:06
• At what point is pi ever not an approximation? I think five places is pretty good.
– yoel
Commented Jun 6, 2012 at 19:08
• @Curiouser, Early authorities had already proposed that the reason the "ה" is not extant in the kri is to hint to the fact that the measurement is in fact an approximation and is actually "missing" some length. The Gr"a is merely showing that there is a neat remez that can show how the number was approximated, that is the approximation is essentially a scaling by the ratio of the kri with the ksiv. There is no "methodology". If it would have been easier to hint to e.g. the difference instead of the scaling factor, perhaps it would have.
– jake
Commented Jun 6, 2012 at 19:48
• This is false. The gra never said this. Commented May 4, 2017 at 14:02
• judaism.stackexchange.com/q/84337/759 Commented Jul 26, 2017 at 13:56

Wikipedia has a set of answers in their article on Approximations of pi. That links to a terrific article on rabbinic approximations of π by Boaz Tsaban and David Garber. Tsaban and Garber summarize as follows (pp. 10-11):

1. The rational-religious approach of Maimonides holds that, since we cannot know the exact values, the Bible tells us that we do not have to worry about this and that is suffices to use the value 3.
2. The mystical approach of [Matityahu Hacohen] Munk contends that 3 was indeed the ration of the circumference to the diameter in King Solomon's temple: This value is used in order to bridge the gap between our world and the "world of truth." For the sake of consistence, the halachic conditions are applied to the suitable regular polygons.
3. The practical approach of R' Shimon Ben Tsemah [who learns from other places in Talmud that they used a more precise version of π] asserts the the rough approximations are used when teaching the students, but, when it comes to practice, the calculations are to be done by the experts.

So to answer your question, if you hold by Munk (I don't know who he is), then it's a miracle. If you hold by Rambam or R' Shimon ben Tsemah, it's an approximation

((Aside: two different psaks come out of this for practical reasons like sukkot - either you use the best mathematical approximation (R' Shimon ben Tsemah) or you use 3 (Rambam and Munk). In order to use 3 as π, you can just measure the perimeter of the interior inscribed regular hexagon.))

• Rabbi Max Munk "three geometry problems in tanach and talmud" (Hebrew), SINAI, 51: 218-227 (Harav Kook Institution, 5722) -- one of the sources used in books.google.co.cr/… -- looks like it's the same Commented Jun 6, 2012 at 23:17

I think that the point is being missed here.

There are not that many places where there is a difference between the written word (k'siv) and the way the word is pronounced (kri). This is especially true where the written word would be pronounced the same way. The reason is generally that neither is quite correct. The "real" word should be some combination.

In this case, the gematria of the written word קוה is 111, while the gematria of the spoken word קו is 106. As the Gra shows, this provides a value of 3.1415.

Everyone seems to be impressed that Archimedes placed pi between 3 1/7 and 3 10/71 around 300 BCE, but that is between 3.1408 and 3.1429. The book of Kings was written about 600 BCE. Mathematics was not advance enough at this time to any person to provide this accuracy. It seems to me to less credulous to believe that there is a divine aspect than to say that it is a coincidence.

• Nice false dichotomy. Commented Jul 17, 2013 at 21:00
• This dichotomy is what I am presented with in discussion with those who do not accept Torah. They tend to attribute most things to coincidences. Your alternative explanation is....? Commented Jul 17, 2013 at 21:49
• @StevenSchulman The problem with this answer is that it seems to go against the Gemara. Commented Jul 17, 2013 at 22:44
• Hi @StevenSchulman and welcome to mi.yodeya! The idea that the true meaning of words with a different k'ri than k'siv is an interesting one. Does it have a written source?
– WAF
Commented Jul 18, 2013 at 14:34
• So it's a miracle? Commented Sep 23, 2013 at 2:55

1 - The Chazon Ish (O.C. 138:4) writes that this is an application of the general principle that "שיעורין הלכה למשה מסיני," meaning that halachic measurements are matters of Divine oral tradition. Thus, the verse (and the Talmud thereon) are not attempting to estimate pi, but rather to teach the halachic value of pi which should be used, as per the halachic tradition. (A novelty of this suggestion is that the הלכה למשה מסיני would be telling us to suspend the true mathematical calculation in favor of an inaccurate one, whereas normally שיעורין הלכה למשה מסיני applies to matters where we would have no other basis for determining the measurement, such as the volume of bone matter to cause impurity or the volume of food to constitute eating.)

2 - The Ein Eliyahu says that the "sea" was a hexagonal shape and therefore the calculation is precise. (With regards to only the diagonal and perimeter, this works out very neatly with a regular hexagon in which each side is 5 cubits, and the diagonal is therefore 10 cubits. However, the Ein Eliyahu seems to not be discussing a regular hexagon, as this does not resolve the issue of calculating the volume, which is what he is discussing. His assertion works out with certain non-regular hexagons. ואכמ"ל.)

3 - The Tiferes Tzvi (R' Tzaddok HaKohen) to Yoreh De'ah 30 says that pi is indeed exactly 3, as the verse and the gemara state, and shame on those who would accept the words of geometrists over the wisdom of our Sages!

There is a very, very full and wonderful essay on this topic (in English, translated from the Ruusian original) which can be found here, but I will quote two paragraphs which will significantly add to what has been discussed here already:

It appears to me that the correction קו/קוה (qava/qav) has not merely numerical meaning. The word קוה (qava) is feminine (in Hebrew the feminine words almost always end with ה) while קו (qav) is masculine. The way the word is spelled is called "masoret"-מסורת and is feminine, the way it is pronounced is called "mickra"- מקרא and is masculine. On the other side, in the pair circle-diameter, the circle represents a feminine, material notion (e.g. the mother Earth) while the straight line represents the masculine, spiritual notion (e.g. the rain that fertilizes the earth). Hence the word קוה (qava) is related to the circle while קו (qav) to the diameter. With this correspondence the verse 7:23 reads "קו (qav) ten cubits from the one brim to the other … and a קוה (qava) of thirty cubits did circle it round about". Thus the ratio of a circle to diameter becomes (30xqava)/(10xqav)=333/106.

Notice that that all objects in the tabernacle where straight. May be this is the reason why Rambam draw the Menorah with straight branches? If in the "heavens", in the spiritual world, there are no curved lines, the circle is perhaps represented there by a polygon. In case of the perimeter of the circle, the hexagon could serve as a model. In case of the area, the dodecagon could be the model. In the first case the perimeter is equal 2∙3∙radius of the surrounding circle; in the second case the area equals 3∙square of the radius, as if π=3. That is why the Sages considered the equality π=3 not as an acceptable approximation but as a reflection of a certain spiritual truth.

There is a lot more material there and it is well worth a look, especially if you are mathematically inclined.

• I think we already have two such answers Commented Nov 21, 2013 at 7:12
• @ShmuelBrin - Read it again.
– user4523
Commented Oct 7, 2014 at 8:31

Perhaps the width of the metal accounts for the discrepancy. The diameter being the outside diameter and the circumference being the inside circumference. Perhaps.
Perhaps as this was a known issue, π = 3 was considered an acceptable standard for illustrative purposes.

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.

1 Kings 7:23 NIV

Outside diameter is 10 (cubits)

Inside circumference is 30 (cubits) Therefore, inside diameter is :
30 ÷ π ≈ 9.55

Outside diameter minus inside diameter :
10 - 9.55 = 0.45 or roughly half a cubit.

This difference is shared equally between the two widths of the material at the two brims.
The width of the material is therefore about a quarter cubit.

It was a handbreadth (טֶ֔פַח) in thickness ...

1 Kings 7:26 NIV

Perhaps the best clue we have as to this :

All these structures, from the outside to the great courtyard and from foundation to eaves (הַטְּפָח֔וֹת), were made of blocks of high-grade stone ...

1 Kings 7:9 NIV

Most modern translations consider this to be the coping.
While no doubt the coping was stone, this is without doubt a reference to the corbelling, which with supporting column, resemble a raised forearm with upward facing palm, hence "hand". Or alternately, a raised forearm with clenched fist. Both having the resemblance.

A common definition of a cubit is the distance between the elbow and the index finger with an outstretched hand.
However, with a clenched fist, a handbreadth, is a quarter the distance between the knuckles and the elbow which squares with the difference between the width of the bowl and its diameter.

This fits but we simply don't know the bible cubit or handbreadth.

However, at the end of the day, this is not instruction from deity, it is a record of what was built by Solomon according to those that measured and recorded it.

It took Solomon thirteen years, however, to complete the construction of his palace. He built ...

1 Kings 7:2 NIV

A record of the work done, no doubt by stretching rope across one of the diameters (watch out for those chords) and then running that rope around one of the circumferences :

He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line (קָוֶה) of thirty cubits to measure around it.

1 Kings 7:23 NIV

Any so inclined ancient artisan could determine that the circumference of a circle was slightly greater than three times its diameter.
Sure enough, this was known long before Solomon.

## Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics.
It dates to around 1550 BC.

The fractional term 256/81 approximates the value of π as being 3.1605..., an error of less than one percent.

https://en.m.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

Not co-incidentally this corresponds with the chronology of the servitude in Egypt.
No doubt this also a great matter of interest some 430 years prior when Joseph was in charge of the pantry for the biblical world.
Calculating the volume of grain in a cylindrical granary, would have been a big deal before, during and after.

There's record of other ancient civilizations grappling with π - it was a known issue.
It's more than reasonable that either the hebrews were aware of the intellectual matter, or the practical matter, or both.

It is what it is.
There's half a dozen ways to slice this and end up in the same place - that π was a known issue that does not derogate from the inspiration of the bible.

## The last laugh.

As π is an irrational number, there is no capacity to represent it numerically, perfectly, then or now.
In that sense, 3 is just as valid as 3.141592653 ... both imperfect.

The scribes knowing this, and knowing the price of papyrus, shouted "3" and knocked off early to hang out at the temple, smiling amongst themselves as they passed the metal sea, they alone knowing the choice they made that allowed them to save time and money but to also not compromise perfection.

Perhaps the width of the metal accounts for the discrepancy. The diameter being the outside diameter and the circumference being the inside circumference. Perhaps.
Perhaps as this was a known issue, π = 3 was considered an acceptable standard for illustrative purposes.

The law of the Lord is perfect,
refreshing the soul.
The statutes of the Lord are trustworthy,
making wise the simple.

Psalm 19:7 NIV

It is the glory of God to conceal a matter; to search out a matter is the glory of kings.

Proverbs 25:2 NIV